INF3410/4411, Fall 2017

Philipp Hä�igerhafliger@ifi.uio.no

Excerpt of Sedra/Smith Chapter 7: Integrated CMOS Ampli�erBasics and Initial Frequency Analysis

Content

Bias Current Steering (book 7.2)

CS con�guration (book 7.3)

CS Low Pass Characteristics (excerpt book 1.2, 1.6)

CG, and cascode con�guration (book 7.4-7.5)

Improved Current Mirrors/Sources (book 7.6)

Content

Bias Current Steering (book 7.2)

CS con�guration (book 7.3)

CS Low Pass Characteristics (excerpt book 1.2, 1.6)

CG, and cascode con�guration (book 7.4-7.5)

Improved Current Mirrors/Sources (book 7.6)

CMOS Current Steering

Computing Rref CMOS Current Steering

(live on whitebord)

Wrong: using the small signal input resistance1

gmfor a desired ID

Correct: Use large signal model!!!

Dynamic Input Currents, Current Ampli�er

Content

Bias Current Steering (book 7.2)

CS con�guration (book 7.3)

CS Low Pass Characteristics (excerpt book 1.2, 1.6)

CG, and cascode con�guration (book 7.4-7.5)

Improved Current Mirrors/Sources (book 7.6)

Intrinsic Gain from Small Signal

Intrinsic Gain vs Bias Current

CS Ampli�er with Current-Source Load

CS Ampli�er Analysis

Content

Bias Current Steering (book 7.2)

CS con�guration (book 7.3)

CS Low Pass Characteristics (excerpt book 1.2, 1.6)

CG, and cascode con�guration (book 7.4-7.5)

Improved Current Mirrors/Sources (book 7.6)

A �rst look at high frequency behaviour

What happens to an ampli�er gain for input signals of di�erentspectra, e.g. when the input is a high frequency/low frequencysignal?

Gain in Decibels

Av =vO

vI

Ap =pO

pL=

vO iO

vI iI= AvAi

Ap,dB = 10 log10 Ap [dB]

Av ,dB = 20 log10 Av [dB]

Arbitrary Signals

Any arbitrary signal can beexpressed as an (in�nite) sum or(in�nite) integral of sine wavesignals of di�erent frequenciesand phases by means of theinverse Fourier transform:

vS(t) =1

2π

∫ ∞−∞

v̂s(ω)e iωtdω

Where the Fourrier transform v̂s is a complex number. How doesthe above equation represent a 'sum of sine waves'?

Cyclic Signals

Cyclic signals can be expressed asdiscreet sums of sine wavesignals, more precisely withharmonics of the cycle frequency.

Example: Approximating Square Wave

Here is how a square wave can beapproximated with the sum of 3sine waves.

v(t) =4V

π

(sinω0t +

1

3sin 3ω0t +

1

5sin 5ω0t + ...

)(1.2)

Spectrum of Square Wave

Thus the spectrum of a square wave (or any cyclic signal) isnon-zero at the harmonics (multiples of the fundamental frequency)only. It can thus be expressed as a sum.

Spectrum of Arbitrary Function

whereas the spectrum of a non-cyclic function can be non-zero forany frequency, e.g. the function in �gure 1.3.

Transfer FunctionA linear circuit network that receives an input signal and producesan output signal is a linear �lter and can be completelycharacterized by it's transfer function T (ω) where ω is a realnumber representing the frequency of the spectrum and T is acomplex number.Per de�nition, T is the transfer function of a �lter if the spectrumof the outout signal Y (ω) is equal to the product of the inputsignal spectrum X (ω) and T (ω)

Y (ω) = T (ω)X (ω) ⇔ Y (ω)

X (ω)= T (ω)

So the magnitude of T (ω) represents a gain and the anglerepresents a phase shift that is applied to the input signal spectrumat each frequency.In the context of circuits one normally writes T (jω) = T (s) whichis the same as above function T (ω) as the term jω consistentlyappears in T when it is derived from analysing circuits and it'ssimpler to just write s in the derivation.

Bode Plot

Transfer functions can be fully described by Bode plots. Forexample:1st Order Low-Pass Filter

Impedances of Linear Circuit Elements

A transfer function of a linear circuit can be conveniently derived bysimply applying Ohm's and Kircho�'s laws to the compleximpedances/admittances of the circuit elements.impedance hspace1cm admittanceresistor: R 1

R

capacitor: 1

sCsC

inductor: sL 1

sL

Ideal linearly dependent sources (e.g. the id = gmvgs sources insmall signal models of FETs) are left as they are.Finally solve for vo

vi

Transfer Function Derivation Variants

The transfer function T (s) of a linear �lter is

I the Laplace transform of its impulse reponse h(t). (Left toFYS3220)

I the Laplace transform of the di�erential equation describingthe I/O realtionship that is then solved for Vout(s)

Vin(s). (Left to

FYS3220)

I The circuit diagram solved quite normally for Vout(s)Vin(s)

by

putting in impedances Z (s) for all linear elements according tothe simple rules on the previous slide. (This lecture!)

Transfer Function

Transfer functions T (s) for linear electronic circuits can be writtenas dividing two polynomials of s.

T (s) =a0 + a1s + ...+ ams

m

1 + b1s + ...+ bnsn

T (s) is often written as products of �rst order terms in bothnominator and denominator in the following root form, which isconveniently showing some properties of the Bode-plots. More ofthat later.

T (s) = a0(1 + s

z1)(1 + s

z2)...(1 + s

zm)

(1 + sω1

)(1 + sω2

)...(1 + sωn

)

Example: CS ampli�er with capacitive load

Content

Bias Current Steering (book 7.2)

CS con�guration (book 7.3)

CS Low Pass Characteristics (excerpt book 1.2, 1.6)

CG, and cascode con�guration (book 7.4-7.5)

Improved Current Mirrors/Sources (book 7.6)

CG ampli�er revisited

Rin of a CG ampli�er

vx = (ix + gmvgs)ro + ixRL ⇒ Rin ≈1

gm+

RL

gmro(7.54)

Important step: vgs = −vx

Rout of a CG ampli�er

vx = (ix − gmvgs)ro + ixRS ⇒ Rout ≈ ro + (gmro)Rs (7.58)

CS with source degeneration no RL

Good as current source, e.g in current mirror, but not so good asampli�er.

RO ≈ (1 + gmRS)ro

gm → g ′m =gm

1 + gmRs

Much higher output resistance. Still nonet-increase in gain!

CS with source degeneration with RL

Av = g ′mRORL

RL + RO

≈ gmroRL

RL + (1 + gmRS)ro

... and more degradation to Av due to load RL!

Cascode Ampli�er

Can be looked upon as a CS and CG in series resulting in a intrinsic

combined gain A = gm1ro1gm2ro2 (i.e. with a large load resistance),or a circuit where the CS serves as high quality voltage controlledcurrent source delivering id ≈ gm1vi and the CG bu�ers thatcurrent to a high output resistance ≈ gm2ro2ro1.

Derivation of intrinsic cascode gain

Cascode Ampli�er with In�nite Load Resistance

Folded Cascode with In�nite Load Resistance

Cascode Ampli�er with Finite Load Resistance (1/2)

The load RL must be of equal magnitude as RO to get the bene�tof the increased gain AV ! So here with a simple pFET we are backto square one.

Cascode Ampli�er with Finite Load Resistance (2/2)

Better: employ a cascoded current source.

Dependency of Gain on RL

Content

Bias Current Steering (book 7.2)

CS con�guration (book 7.3)

CS Low Pass Characteristics (excerpt book 1.2, 1.6)

CG, and cascode con�guration (book 7.4-7.5)

Improved Current Mirrors/Sources (book 7.6)

Cascode Current Mirror

Increased output impedance, but quite a bit of output voltageheadroom necessary ...

Modi�ed Wilson MOS Current Mirror

A CD-CS Ampli�er

Larger bandwidth than simple CS ampli�er(explained later in chapter 9).